Calculating statistical moments of the response of mechanical systems in the presence of various uncertainties still remains one of the main topics in stochastic mechanics. To mitigate the “curse of dimensionality” of the existing direct numerical integration methods, the bivariate dimension-reduction method (BDRM) can be invoked. The original BDRM using the Gauss-Hermite quadrature (GHQ) rule, however, is still computationally demanding for the expensive-to-evaluate models. In this paper, three different two-dimensional Gaussian-weighted integration schemes are proposed to improve the computational efficiency of the original BDRM. First, the analytical solution of a nine-point scheme is derived based on a symmetrical sampling strategy and the concept of algebraic accuracy, where the algebraic accuracy is designed to be five. Similarly, two novel thirteen-point schemes on the basis of the nine-point one are then presented, in which their algebraic accuracy is up to seven. In conjunction with the GHQ rules for one-dimensional integrals, three different BDRMs are formulated accordingly. Four numerical examples demonstrate the advantages of the proposed methods for evaluating the first-four statistical moments of the response of stochastic mechanical systems, in comparison to the original BDRM.